Münster J. of Math. 6 (2013), 583–594
urn:nbn:de:hbz:6-55309456692
Münster Journal of Mathematics
c Münster J. of Math. 2013
Z-stability, finite dimensional tracial
boundaries and continuous rank functions
Bhishan Jacelon
(Communicated by Joachim Cuntz)
Abstract. We observe that a recent theorem of Sato, Toms–White–Winter and Kirchberg–
Rørdam also holds for certain nonunital C∗ -algebras. Namely, we show that an algebraically
simple, separable, nuclear, nonelementary C∗ -algebra with strict comparison, whose cone
of densely finite traces has as a base a Choquet simplex with compact, finite dimensional
extreme boundary, and which admits a continuous rank function, tensorially absorbs the
Jiang–Su algebra Z.
1. Introduction
As part of the project of investigating regularity properties of (simple, separable, unital, nonelementary, nuclear) C∗ -algebras, there has been recent
progress in showing that, in certain contexts, tensorial absorption of the Jiang–
Su algebra Z (“Z-stability”) is an automatic consequence of strict comparison
of positive elements. This was proved by Matui and Sato in [16] in the case
of finitely many extremal tracial states, and later extended independently by
Sato [26], Toms, White and Winter [30] and Kirchberg and Rørdam [13] to include C∗ -algebras with compact, finite dimensional extreme tracial boundary.
In this note, we point out that this more general result also holds for algebraically simple C∗ algebras with compact tracial state space (and hence for
those which admit a continuous rank function, in the sense of [8]). We make
use of Nawata’s extension [17] of Matui and Sato’s techniques to the nonunital
setting, of the language of “uniformly tracially large” order zero maps developed in [30], and also of the philosophy of [27], namely that algebraically simple
C∗ -algebras can often be studied using the machinery developed for their unital
cousins.
After establishing notation and tools in Section 2, we present the main
theorem in Section 3 (following the exposition of [30] and [26]). We then show
in Section 4 that for C∗ -algebras with a continuous rank function (in particular,
Research supported by the ERC through AdG 267079.
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Bhishan Jacelon
those that have strict comparison, almost divisibility and stable rank one), the
assumption of compactness of the tracial state space can be removed.
2. Preliminaries
In this section, we record the background results and terminology needed
for the rest of the article.
Cones and traces. If C is a (pointed) convex cone in a locally convex Hausdorff
space, a base of C is a convex subset X of C such that y ∈ C if and only if
y = αx for unique α ≥ 0 and x ∈ X. It is straightforward to show that, if X
is a compact base of C, then the projection C \ {0} → X is continuous, and
(at least if one can choose seminorms defining the ambient topology to all be
strictly positive on C \ {0}) every continuous affine functional on X extends
uniquely to a continuous linear functional on C. Finally, a compact base X
is a Choquet simplex precisely when C is a vector lattice. (See [1] or [21] for
details about cones and simplices.)
Let A be a C∗ -algebra. We denote by Ped(A) the minimal dense ideal of
A (see [20, Chap. 5]), by T (A) the cone of densely finite lower semicontinuous
traces on A (regarded as a subset of the dual of Ped(A) and equipped with
the corresponding weak∗ -topology), and by T1 (A) the space of tracial states
on A, i.e. those elements of T (A) of norm 1. Note that, if A is simple, then
every nonzero element of T (A) is faithful, so in this case any compact base
of T (A) has the extension property mentioned in the previous paragraph. In
particular, this is the case for the following (which are compact bases of T (A)
under the stated hypotheses):
• T1 (A), whenever it is compact and A has no unbounded traces, and
• Ta7→1 (A) := {τ ∈ T (A) | τ (a) = 1}, whenever a ∈ Ped(A)+ is full (see
[29, Prop. 3.4]).
Since T (A) is a lattice (see [18, Cor. 3.3] and [19, Thm. 3.1], also [9, Thm. 3.3]),
any compact base of T (A) is also a Choquet simplex.
For separable, simple A, we define
Aff + (T (A)) := {f : T (A) → [0, ∞) | f linear, continuous, f (τ ) > 0 for τ 6= 0}
SAff + (T (A)) := {f : T (A) → [0, ∞] | fn ↑ f pointwise, fn ∈ Aff + (T (A))}.
Note that the norm map k · k : T (A) → [0, ∞], which we will sometimes denote
by ωA , is an element of SAff + (T (A)). Moreover, if A has no unbounded traces,
then T1 (A) is compact precisely when ωA is continuous.
S
+
The Cuntz semigroup. The Cuntz relation on M∞ (A)+ := ∞
is
k=1 Mk (A)
∗
defined by a - b if ka − vn bvn k → 0 for some vn , and ∼ denotes the equivalence relation that symmetrizes it. The Cuntz semigroup W (A) is the set
of equivalence classes [a] of elements a ∈ M∞ (A)+ . It is a positively ordered
abelian monoid under the addition [a] + [b] = [diag(a, b)] and the partial order
[a] ≤ [b] if a - b. We refer the reader to [2] for more information.
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Every τ ∈ T (A) gives rise to a lower semicontinuous functional dτ : W (A) →
[0, ∞], defined by dτ ([a]) = limn→∞ τ (a1/n ). (Lower semicontinuity of dτ
means precisely that dτ ([a]) = supε>0 dτ ([(a − ε)+ ]) for every a ∈ M∞ (A)+ ,
where (a − ε)+ is defined to be the element of C∗ (a) (in fact, of Ped(C∗ (a)))
corresponding under functional calculus to f (t) := max{0, t − ε}.)
As in [8] (see also [22, Rem. 6.0.4] and [1, Cor. I.1.4]), when A is simple and
separable we define
ι : W (A) → SAff + (T (A)) by ι([a])(τ ) := dτ (a) for [a] ∈ W (A), τ ∈ T (A).
(Note that if e ∈ A is strictly positive, then ι([e]) = ωA , i.e. dτ (e) = kτ k for
every τ ∈ T (A).)
An exact C∗ -algebra A is said to have strict comparison if, whenever a, b ∈
M∞ (A)+ with dτ (a) < dτ (b) for every τ ∈ T (A) such that dτ (b) = 1, then
a - b. We say that A (or W (A)) has almost divisibility if for every y ∈ W (A)
and m ∈ N there exists x ∈ W (A) such that mx ≤ y ≤ (m + 1)x. By
results of Rørdam [23], every separable (exact) Z-stable C∗ -algebra has both
of these properties (see [33, Prop. 3.7], the proof of which works equally well
for nonunital C∗ -algebras, and also [9, Prop. 6.2]).
Tracially large order zero maps. Recall that a completely positive (c.p.) map
has order zero if it preserves orthogonality (see [34]). Given a C∗ -algebra A,
we denote by A∞ the central sequence algebra A∞ ∩ A′ := ℓ∞ (A)/c0 (A) ∩ A′ ,
where A is embedded as the set of constant sequences. Since order zero maps
on Mk := Mk (C), k ∈ N, are projective (by results of Loring [15]; see also [30,
Lemma 2.1]), every completely positive and contractive (c.p.c.) order zero map
∞
Φ : Mk → A∞ lifts to a c.p.c. order zero map (ϕn )∞
n=1 : Mk → ℓ (A), where
each ϕn is a c.p.c. order zero map Mk → A. As in [30], if A is separable and
T1 (A) is a nonempty base of T (A), such a Φ is said to be uniformly tracially
large if
lim
inf τ (ϕn (1k )) = 1
n→∞ τ ∈T1 (A)
for some (hence every) such lifting (ϕn )∞
n=1 .
For separable, simple, unital, infinite-dimensional, nuclear C∗ -algebras with
finitely many extremal tracial states, the existence of uniformly tracially large
order zero maps is guaranteed by [16, Lemma 3.3]. In general, one has the
following.
Theorem 2.1 (Matui–Sato; Nawata). Let A be a separable, simple, nonelementary, nuclear C ∗ -algebra with strict comparison, such that every trace on A
is bounded and T1 (A) is nonempty and compact. Suppose that for every k ∈ N,
there is a uniformly tracially large c.p.c. order zero map Φ : Mk → A∞ . Then
A is Z-stable.
In the unital case, strict comparison is used in [16] to prove that every c.p.
map A → A can be excised in small central sequences, and hence that A has
property (SI) (see [16, Def. 2.1] and [16, Def. 4.1] for the relevant definitions).
As presented in [16], Φ plays a role in this part of the argument; namely, to
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Bhishan Jacelon
ensure that the set of excisable c.p. maps is closed under finite sums. However,
it is demonstrated in [13] that this is unnecessary (compare [13, Prop. 5.9] with
[16, Lemma 3.1]). We are grateful to the referee for pointing this out, and for
the observation that Φ is only needed to show that Z-stability can be deduced
from property (SI): when (SI) is applied to certain central sequences obtained
from Φ, one exhibits in A∞ the order zero relations for the dimension drop
algebra Zk,k+1 described in [24, Prop. 5.1]; by [31, Prop. 2.2], this implies
Z-stability.
If A does not necessarily have a unit, then it is shown in [17] that the
appropriate nonunital version of property (SI) holds (see [17, Sec. 5]). The
Zk,k+1 -relations are then witnessed not in A∞ but in the quotient F (A) of A∞
by the left annihilator of A, first defined in [12]; from this, Z-stability follows
(see [12, Prop. 4.11] and also [17, Prop. 5.1]).
Algebraically simple C ∗ -algebras. A simple C∗ -algebra is algebraically simple
precisely when A = Ped(A) (in particular, if A is unital). It follows from [20,
Prop. 5.6.2], [7, Lemma 5.6] and Brown’s theorem [4] that every simple, σunital C∗ -algebra is stably isomorphic to an algebraically simple C∗ -algebra.
Such algebras are tracially well behaved in the following sense.
Proposition 2.2. The following hold for a separable algebraically simple C ∗ algebra A.
(i) Every trace on A is bounded (so T1 (A) is a base of T (A)).
(ii) The weak∗ -closure of T1 (A) in A∗ does not contain 0; equivalently,
inf
τ ∈T1 (A)
τ (a) > 0 for every a ∈ A+ \ {0}.
(iii) Suppose that T1 (A) is nonempty and compact. For any strictly positive
continuous affine functional f on T1 (A) and for any ε > 0, there exists
a ∈ A+ with τ (a) = f (τ ) for every τ ∈ T1 (A) and kak < kf k + ε.
(iv) Let A be as in (iii). Then
lim
sup |τ (en a) − τ (en )τ (a)| = 0
n→∞ τ ∈T (A)
1
for any central sequence (en )∞
n=1 ∈ A∞ and any a ∈ A.
(v) Suppose that A is as in (iii) and is also nuclear. Then for every closed
subset X ∈ ∂e (T1 (A)) and mutually orthogonal positive functions f1 ,
. . . , fN ∈ C(X) of norm ≤ 1, there exist central sequences (ai,n )∞
n=1
of positive contractions in A such that
lim sup |τ (ai,n ) − fi (τ )| = 0 and lim kai,n aj,n k = 0 for i 6= j.
n→∞ τ ∈X
n→∞
Proof. See [27, Sec. 2] for the proofs of the first two assertions. For the remaining three, the assumption that T1 (A) is compact ensures that every continuous
affine functional on T1 (A) extends to a continuous linear functional on T (A).
(Moreover, by [1, Thm. II.3.12], if X ⊂ ∂e (T1 (A)) is closed, then every continuous function on X extends to a continuous affine functional on T1 (A) of the
same norm.) Then, (iii) is proved exactly as in [14, Thm. 9.3]; Lin’s theorem
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is stated for simple, unital C∗ -algebras, but this assumption is only needed to
invoke [7, Cor. 6.4], which holds for algebraically simple C∗ -algebras in general. The final two statements are the same as [26, Lemma 4.2 (i)] and [26,
Lemma 4.2 (ii)] respectively, and are proved in exactly the same way. Note in
particular that only when appealing to [26, Cor. 3.3] or [26, Prop. 4.1] is a unit
being used. However, the proof of the former works just as well for nonunital
algebras (with the unitaries taken in the unitization), and the latter holds for
algebraically simple algebras in light of (iii).
Remark 2.3. It can be shown from the results of [3] (see also [11, Prop. 5.3])
that for every infinite-dimensional metrizable Choquet simplex X with compact extreme boundary ∂e (X), there exists an algebraically simple AF algebra
A such that T (A) has a base affinely homeomorphic to X, yet T1 (A) is not
compact.
3. The main theorem
Theorem 3.1. Let A be a separable, algebraically simple, nonelementary, nuclear C ∗ -algebra and suppose that T1 (A) and ∂e (T1 (A)) are nonempty and compact, and ∂e (T1 (A)) has finite covering dimension. Then A admits a uniformly
tracially large order zero map Φ : Mk → A∞ for every k ∈ N. In particular,
such C ∗ -algebras are Z-stable whenever they have strict comparison.
We will use the following method of building up large order zero maps from
Mk . (See [13, Lemma 7.6], also [26, Cor. 2.3].)
Proposition 3.2. Let B be a C ∗ -algebra, k ≥ 2, and ϕ1 , . . . , ϕN : Mk → B
c.p.c. order zero maps whose images commute, and such that kϕ1 (1k ) + · · · +
ϕN (1k )k ≤ 1. Then there exists a c.p.c. order zero map ψ : Mk → B with
ψ(1k ) = ϕ1 (1k ) + · · · + ϕN (1k ).
It will moreover be sufficient to verify the conditions of the proposition
approximately. That is, if A is separable and there exists N ∈ N such that, for
every finite subset F ⊂ A and every tolerance ε > 0, there are c.p. maps ϕl,n ,
∞
are c.p. order zero maps
l = 1, . . . , N , n ∈ N, such that (ϕl,n )∞
n=1 : Mk → A
with commuting images and satisfy
N
X
lim sup ϕl,n (1k ) ≤ 1,
(3.1)
n→∞
l=1
lim inf
inf
n→∞ τ ∈T1 (A)
N
X
τ (ϕl,n (1k )) ≥ 1 − ε and
l=1
lim sup k[ϕl,n (x), a]k ≤ εkxk for x ∈ Mk , a ∈ F, l = 1, . . . , N,
n→∞
then, by a diagonal argument and Proposition 3.2, there exists a uniformly
tracially large order zero map ψ : Mk → A∞ .
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Bhishan Jacelon
Proof of Theorem 3.1. The proof is exactly the same as in [30, Lemma 3.5]
(for the zero dimensional case) and [26, Prop. 5.1] (for the general case), so we
will try to just give a brief outline. Most of the work needed to carry these
arguments over to nonunital C∗ -algebras is either contained in Proposition 2.2
or has already been done by Nawata in [17]. In particular, we will appeal when
necessary to [17, Prop. 5.3], which says the following: if A is a C∗ -algebra with
a countable approximate unit (hn )∞
n=1 and with T1 (A) nonempty and compact,
then
(3.2)
lim
sup |τ (hn gn ) − τ (gn )| = 0
n→∞ τ ∈T (A)
1
for any sequence (gn )∞
n=1 of positive contractions in (the unitization of) A.
Finally, by dim we mean the topological covering dimension, which for separable metric spaces coincides with the (small) inductive dimension (see [10]).
Fix a finite set F ⊂ A and a tolerance ε > 0, and suppose that X ⊂
∂e (T1 (A)) is closed. By the argument of [16, Lemma 3.3] (see also [17, Lemma
5.9] and [30, Lemma 2.9]), for every τ ∈ ∂e (T1 (A)) there exists a c.p.c. order
zero map ϕτ : Mk → A such that k[ϕτ (x), a]k < εkxk for every x ∈ Mk , a ∈ F
and τ (ϕτ (1k )) > 1 − ε. The latter inequality holds on an open neighborhood
Uτ of τ in ∂e (T1 (A)), and by compactness there exist τ1 , . . . , τM ∈ ∂e (T1 (A))
such that X is covered by U := {Uτ1 , . . . , UτM }.
The zero-dimensional case. Suppose that dim X = 0. Then there exists a
relatively open finite cover V = {V1 , . . . , VN } of X that refines U and whose
elements are pairwise disjoint closed sets. For each j ∈ {1, . . . , N }, let fj :
X → [0, 1] be continuous with fj |Vj = 1 and fj |Si6=j Vi = 0. By Proposition 2.2(v), there exist central sequences (ai,n )∞
n=1 of positive contractions in
A, i = 1, . . . , N , with
(3.3)
lim sup |τ (ai,n ) − fi (τ )| = 0 and lim kai,n aj,n k = 0 for i 6= j.
n→∞ τ ∈X
n→∞
For each j, let l(j) be such that Vj ⊂ Uτl(j) ∩ X. Define ψn : Mk → A by
PN 1/2
1/2
∞
is c.p.c.
ψn (x) := j=1 aj,n ϕτl(j) (x)aj,n . Then ψ := (ψn )∞
n=1 : Mk → A
are
pairwise
orthogonal).
order zero (since each ϕτl(j) is, and the (aj,n )∞
n=1
Moreover, for x ∈ Mk and a ∈ F we have
lim sup k[ψn (x), a]k ≤ max k[ϕτl(j) (x), a]k < εkxk
j∈1,...,N
n→∞
(once again using pairwise orthogonality of the (aj,n )∞
n=1 ), and for each j and
ρ ∈ Vj , we have ρ(ϕτl(j) (1k )) > 1 − ε (since Vj ⊂ Uτl(j) ). From this latter inequality, together with Proposition 2.2(iv) and the fact that, by (3.3),
limn inf ρ∈Vj ρ(aj,n ) = inf ρ∈Vj fj (ρ) = 1, it follows that
lim inf inf ρ(ψn (1k )) ≥ 1 − ε.
n→∞ ρ∈X
If X = ∂e (T1 (A)), then by the Krein–Milman theorem, the conditions of (3.1)
are satisfied.
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The general case. The argument is by induction on the dimension of closed
subsets of ∂e (T1 (A)). Let c be an integer with 1 ≤ c ≤ d := dim ∂e (T1 (A)) and
let X ⊂ ∂e (T1 (A)) be closed with dim(X) = c. Taking a suitable refinement
V = {V1 , . . . , VN } of U, and arguing as in [26], we obtain the following data:
• a closed subset X0 of X of dimension ≤ c − 1 (this uses the notion of
inductive dimension; X0 is the union of boundaries of elements of V);
• inductively, c.p.c. order zero maps ψl,n : Mk → A, l = 0, . . . , c − 1, n ∈
N such that
c−1
X
lim sup ψl,n (1k ) ≤ 1,
(3.4)
n→∞ l=0
lim k[ψl,n (x), a]k = 0 for a ∈ A, x ∈ Mk ,
n→∞
lim
inf
n→∞ τ ∈X0
c−1
X
τ (ψl,n (1k )) = 1, and
l=0
lim k[ψl,n (x), ψm,n (y)]k = 0 for x, y ∈ Mk , l 6= m;
n→∞
• for each n ∈ N, a relatively open subset W0,n ⊂ X containing X0 such
that
(3.5)
inf
τ ∈W0,n
c−1
X
τ (ψl,n (1k )) > 1 − εn , εn → 0;
l=0
• pairwise disjoint relatively open subsets W1 , . . . , WN ⊂ X (obtained in
a straightforward manner as a refinement of V) such that Wj ⊂ Uτl(j)
for every j and some l(j), and such that {W1 , . . . , WN } covers X \ X0
(in particular, Wn := {W0,n , W1 , . . . , WN } covers X for every n ∈ N).
Take a partition of unity {f0,n }N
i=0 ⊂ C(X) subordinate to Wn , and use Proposition 2.2(v) to obtain central sequences (ai,n,m )∞
m=1 of positive contractions
in A, i = 1, . . . , N , such that
(3.6) lim sup |τ (ai,n,m ) − fi,n (τ )| = 0 and lim kai,n,m aj,n,m k = 0, i 6= j.
m→∞ τ ∈X
m→∞
Fix an approximate unit (hm )∞
m=1 for A. For every n ∈ N, choose a strictly
increasing sequence (kn,m )∞
m=1 of positive integers such that (taking the supremum over p ≥ kn,m , 1 ≤ j ≤ N , 0 ≤ l ≤ c − 1, 1 ≤ m′ ≤ m and x in the unit
ball of Mk ):
o
n
1
1/2
(3.7)
sup max kh1/2
.
p aj,n,m′ − aj,n,m′ k, khp ψl,n (x) − ψl,n (x)k <
′
m
p,j,l,m ,x
Set en,m := hkn,m , so that (en,m )∞
m=1 is an approximate unit for A, and in
particular represents an element of A∞ . For each n ∈ N, since the (ai,n,m )∞
m=1
are pairwise orthogonal, we may choose positive contractions a0,n,m , m ∈ N,
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Bhishan Jacelon
in A such that
(a0,n,m )∞
m=1
∞
N
X
1/2
1/2
ai,n,m en,m
= en,m 1 −
∈ A∞ .
m=1
i=1
Then, by (3.7), (a0,n,m )∞
commutes with (ai,n,m )∞
m=1 in A∞ for i = 1, . . . , N .
PN m=1
Since f0,n = 1 − i=1 fi,n , we also have, by (3.2) and (3.6), that
lim sup |τ (a0,n,m ) − f0,n (τ )| = 0.
m→∞ τ ∈X
By a diagonal argument, using separability of A, (3.7) and, for the last assertion, Proposition 2.2(iv), we can find a subsequence (mn )∞
n=1 of positive
integers such that:
• (aj,n,mn )∞
n=1 , j = 0, 1, . . . , N , are central sequences whose sum in A∞
is a contraction;
• for j = 1, . . . , N , the (aj,n,mn )∞
n=1 are pairwise orthogonal and all
in
A
commute with (a0,n,mn )∞
∞;
n=1
∞
commutes
with
(ψ
• (aj,n,mn )∞
l,n (x))n=1 in A∞ for j = 0, 1, . . . , N ,
n=1
l = 0, . . . , c − 1 and x ∈ Mk ;
• limn supτ ∈X |τ (aj,n,mn ) − fj,n (τ )| = 0 for j = 0, 1, . . . , N ;
• limn supτ ∈X |τ (a0,n,mn ψl,n (x)) − τ (a0,n,mn )τ (ψl,n (x))| = 0 for l = 0,
. . . , c − 1 and x ∈ Mk .
Then, using these properties together with (3.4) and (3.5) and arguing just as
in the zero dimensional case, the maps ϕl,n : Mk → A, l = 0, . . . , c, defined by
1/2
1/2
ϕl,n (x) := a0,n,mn ψl,n (x)a0,n,mn for l = 0, . . . , c − 1, and
ϕc,n (x) :=
N
X
1/2
1/2
aj,n,mn ϕτl(j) (x)aj,n,mn ,
j=1
can be shown to satisfy (3.1) (for X). Thus, by induction, there exists a
uniformly tracially large order zero map Mk → A∞ . The statement about
Z-stability then follows from Theorem 2.1.
Note that, as defined, uniformly tracially large order zero maps cannot be
used to access algebraically simple C∗ -algebras with non-compact tracial state
space.
Proposition 3.3. Suppose that A is separable and algebraically simple, and
that for some k ∈ N, there exists a uniformly tracially large order zero map
Φ : Mk → A∞ . Then T1 (A) is compact.
Proof. Let a be a nonzero element of A+ and choose (by (ii) of Proposition 2.2)
M > 0 such that ωA (σ) := kσk < M for every σ ∈ Ta7→1 . Let (ϕn )∞
n=1 be a
lift of Φ to c.p.c. order zero maps Mk → A, and let (εn )∞
n=1 be a decreasing
sequence of positive real numbers converging to zero. For every n ∈ N, let
mn ∈ N be large enough so that an := ϕmn (1k ) satisfies
|τ (an ) − 1| < εn for every τ ∈ T1 (A),
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Z-stability, traces and continuous rank functions
591
hence
|σ(an ) − kσk| < M εn for every σ ∈ Ta7→1 .
Thus ωA is a uniform limit of continuous functions, so is continuous on Ta7→1 .
Since Ta7→1 is a compact base of T (A), it follows that ωA is continuous on all
of T (A), and hence that T1 (A) is compact.
4. Continuous rank functions
In this section, we consider circumstances under which we may drop the
assumption of compactness of the tracial state space in the statement of Theorem 3.1. Let us say that a simple, separable C∗ -algebra A with T (A) 6= {0}
has a continuous rank function if ι(W (A)) ∩ Aff + (T (A)) 6= ∅. This condition
is trivially satisfied if A contains a projection in its stabilization, or if T1 (A) is
a compact base of T (A). Conversely, the following holds.
Lemma 4.1. If A is separable, algebraically simple and has a continuous rank
function, then there exist k ∈ N and a hereditary subalgebra B of Mk (A) such
that T1 (B) is compact.
Proof. Let b ∈ Mk (A)+ be such that ι([b]) is continuous. Then the hereditary
subalgebra b(Mk (A))b =: B generated by b in Mk (A) is also algebraically simple. Moreover, the restriction map ρB : T (A) → T (B) is a linear isomorphism
(see for example [7, Prop. 4.7]), so ωB = ι([b]) ◦ ρ−1
B is continuous. Hence,
T1 (B) is compact.
Lemma 4.2. If A is separable, simple and has almost divisibility, strict comparison and stable rank one (and T (A) 6= {0}), then there exists an algebraically simple hereditary subalgebra B of A that has a continuous rank function.
Proof. Let B be a (nonzero) hereditary subalgebra of A contained in Ped(A),
so that B is algebraically simple. Note that B inherits stable rank one, strict
comparison and almost divisibility (see also the proof of [27, Cor. 8.7]). Then,
arguing exactly as in [2, Thm. 5.22], one can show that ι(W (B)) contains all
of Aff + (T (B)). Note that one has to work not with T1 (B) but with Ta7→1 (B)
for some fixed a ∈ B + \ {0}, and make use of (ii) and a version of (iii) of
Proposition 2.2, namely:
• sup{kτ k | τ ∈ Ta7→1 (B)} < ∞, and
• every strictly positive continuous affine functional on Ta7→1 (B) is represented by some positive element of B (in a norm-controlled way).
Briefly, the idea is as follows.
Almost divisibility is used to show that, for any ε > 0 and f ∈ Aff + (T (B)),
there exists some b ∈ M∞ (B)+ such that ι([b]) is uniformly close to f on
Ta7→1 (B), within ε (see [2, Lemma 5.20]). Then, for a given g ∈ Aff + (T (B)),
one can write g as the pointwise supremum of an increasing sequence (fn ) ⊂
Aff + (T (B)) in such a way that, taking bn ∈ M∞ (B)+ with ι([bn ]) approximating fn , one has dτ ([bn ]) < dτ ([bn+1 ]) for every τ ∈ Ta7→1 (B) (see [2,
Münster Journal of Mathematics Vol. 6 (2013), 583–594
592
Bhishan Jacelon
Lemma 5.21]). The sequence ([bn ])∞
n=1 is increasing (by strict comparison) and
bounded (since ι([bn ]) converges pointwise to g). Therefore, since B has stable
rank one, [5, Thm. 4.4] (see also [2, Thm. 5.15]) implies that [b] := sup[bn ]
exists in W (B) and satisfies ι([b]) = g.
(Note that it is important for the continuous rank function to be induced
by an element b of some Mk (B), rather than B ⊗ K, so that b generates an
algebraically simple hereditary subalgebra. That is why we work with W (B)
rather than Cu(B) ∼
= W (B ⊗ K), even though (see [6]) suprema of increasing
sequences exist in Cu without the assumption of stable rank one.)
Corollary 4.3. Let A be a simple, separable, nuclear, nonelementary C ∗ algebra with strict comparison, whose cone of densely finite traces has as a
base a Choquet simplex with compact, finite dimensional extreme boundary.
Suppose also that A has an algebraically simple hereditary subalgebra with a
continuous rank function (for example, if A has almost divisibility and stable
rank one). Then A is Z-stable.
Proof. By Lemma 4.1 and Brown’s theorem [4], there is an algebraically simple
C∗ -algebra B that is stably isomorphic to A and satisfies the hypotheses of
Theorem 3.1. The result then follows because Z-stability is preserved under
stable isomorphism [32].
Remark 4.4. Simple, nonelementary, approximately subhomogeneous C∗ algebras with no dimension growth have strict comparison and almost divisible
Cuntz semigroup (see [28, Cor. 5.9] and [29, Cor. 7.2]), and also have stable
rank one (see [25, Thm. 3.2]). Therefore, such algebras are Z-stable whenever
the assumption on the tracial cone also holds. This has already been shown in
[27, Cor. 9.2], where no such assumption is necessary, and where “no dimension
growth” can be replaced by “slow dimension growth”.
Acknowledgments
I am grateful to Wilhelm Winter, Karen Strung, Aaron Tikuisis and Stuart
White for many helpful discussions.
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Received January 16, 2013; accepted April 15, 2013
Bhishan Jacelon
Westfälische Wilhelms-Universität Münster, Mathematisches Institut,
Einsteinstr. 62, D-48149 Münster, Germany
E-mail: [email protected]
Münster Journal of Mathematics Vol. 6 (2013), 583–594